Irreducible unitary representations of the group of diffeomorphisms of

827

J. Math. K yoto Univ. (JMKYAZ) 33-3 (1993) 827-864

Irreducible unitary representations of the group of diffeomorphisms of a non-compact manifold By Takeshi HIRAI

Introduction Let M be a connected paracompact C -manifold an d G=Diffo(M) the group of all diffeomorphisms of M with compact supports. In this paper, we construct new series of irreducible unitary representations ( =IURs) of G, for a non-compact M , by using a certain kind o f product measures on X = rizeNM„ M = M , a n d IURs of the infinite symmetric group a o of all finite permutations on the set N of natural numbers. 1. The group G acts on X from the left and the group a. acts from the right through permutations of the coordinates. T he latter produces intertwining operators when we consider representations o f G on /2-spaces of ao -invariant measures on X or tensor products (with respect to some reference vectors) of natural representations of G on the Hilbert spaces /2(M ,). If we can decompose these representations into irreducibles, then we will obtain many different IURs of G . Actually we do not proceed in this direction, but our results obtained here can be viewed, at least from the spirit, as an infinite version of the Weyl's beatiful situation in [18] for finite-dimensional (holomorphic) irreducible reprensentations of the full linear group G '= G L (n , C): for the tensor product (X=1 V , V, V, of natural representation of G' on V = Cn, the symmetric group c of indices {1, 2, •••, N I generates the algebra of its intertwining operators, and thanks to this, there exists a natural correspondence between IURs of c and irreducible representations of G' with Young diagrams of rank N (for any N 1 ) . 2. Let us explain our method and results more e x a c tly . An element (x,),eN of X is called an ordered configuration in M , if the underlying set of points {z},eN is a configuration in M or it has no accumuration points and x, * x , (1 *]). T h e s e t X o f all ordered configurations can be viewed as a principal bundle with a base set Q = X / and fibres a o . We can introduce on ..(2no suitable topology but rather good measurable structures consistent to those on g , and then, to construct IURs of G, we can apply a version of the standard method of associated vector bundles. -

N

N

-

Received M ay 7, 1992.

828

Takeshi Hirai

We proceed as follows. Take a measure p on M , locally equivalent to Lebesgue measures, defined on the family E m of all Lebesgue measurable subsets of M . We call a subset of X of the form E E , E T t m , u n i t a l if .E,'s are mutually disjoint and p (E )> 0 , E,N 111(E,) 11< c o . Two unital product subsets E and F=IL E N F, are said to be cofinal (Notation: E F ) if E,enT ,u(EeF,)< cc. For any fixed E , we denote by 0(E) the a-ring generated by the set of unital product subsets IF; F Then vE(F)=-111A rti(F) can be uniquely extended to a measure on 9R(E) which is as-invariant. This measure LIE is supported by je: vE(A )= vE(An k) for AETZ(E), and we get a Using the measure vE on (X , V (E )) and an quotient measure on IUR H of a., we can construct a unitary representation TE of G, attached to such a datum E - (//: p , E ), by a measurable version of the method of associated vector bundles. The irreducibility o f T E is proved in Theorem 4.1. The equivalence criterion for any pair of such repesentations is given in Theorem 5.2. On the way of proving the latter, we encounter an interesting problem, Problem 5.8, on a series (c,,), J EN of non-negative real numbers satisfying the condition that d, - E,ENc,, >0, e, - E,EN c„, >0, and II,N ch, H JN eJ are unconditionally convergent. (This (c,,),,,EN can be said to be, essentially, a stochastic matrix of infinite size.) We remark that the case of I. M. Gelfand et al. [17] is nothing but the case of principal bundle g-0 F = X 1 ... with the group of all permutations on N and the space of all configurations in M . Moreover groups of diffeomorphisms themselves or their homogeneous spaces are also studied by several mathematicians (e.g., [3], [4], [11]-112D. 3. The paper is organized as follow s. Section 1 is devoted to studying .S2= )?1 - with product measures on X and measurable principal bundles measurable structures determined by fixed unital product subsets E and also the structure of the group Diffo(M). In Section 2, we construct unitary representations TE for E =( / / : p , E ) . In Section 3, we collect several lemmas and propositions which are necessary to study irreducibility and equivalence relation for T E 's• In Section 4, the irreducibility is proved. In Section 5, we X =(//; p , E ), E '=(H '; p , E '). The princiestablish the criterion for TE =' pal part of its proof is to get E ' E b for some Acknowledgem ents. The author expresses his hearty thanks to Prof. A. Hora for his important suggestions and kind discussions in the measure theory. The author is also grateful to Prof. K. Nishiyama who read carefully the first draft of this paper and gave many valuable comments. -

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§ 1 . Product measures

Let M be a C -manifold, and Diff(M) the group of all diffeomorphisms of M . For a gEDiff(M), supp(g) is defined as the closure of the set {p m; gp* p}, and put -

829

Irreducible unitary representations G=Diffo(M)— {gEDiff(M); supp(g) is compact).

(1.1)

We equip G with the natural topology: a sequence gn converges to g if supp (g ) and every supP(gn) are contained in a compact set C and gn, together with all its derivatives, tends to g uniformly on C . In this paper, we study irreducible unitary representations ( = IU R s) of G assuming M to be connected and non-compact. To construct IURs of G, we shall use product measures on the infinite product space X = M =ILENM, with Mi = M fo r i E N , where N denotes the set of natural numbers. Notation. For an open subset U and a subset D of M , we put -

G (U )=D iffo(U ) (the group is connected) , GID={gE G; sup p (g)c D}

Note that GIL, G( U) for an open UOEM. Other important subgroups G(E') and G ((E ')) of G are defined in § 3. 1.1.

Measurable structures on the product space X . L et us fix a

measure ,a on M which is equivalent locally to Lebesgue measures, that each coordinate neighbourhood U cM , d i is given as

is, in

du(P)= wu(P)dPidP2. ••dPn , where (Pi, P2, P n ), n = d im M , is the local coordinates of p , and w u(P ) is a positive measurable function. We assume that p(M)=-00 and further that wu is bounded both from below and from above on every compact subset of U so that p (K )< co for every compact subset K of M . The measure ,a is understood as is defined on the family MAI of all Lebesgue measurable subsets of M. In each component M ,= M of X = IL .N M „ take a measurable subset E,, and put E = IL E N E ,. We call such a subset of X u n ita l p rod u ct su b set if it satisfies the following two conditions: .• * ,

(UPS1)

(UPS2)

Nlii(E1) 11 0 (iE N ); -

E , (iE N ) are mutually disjoint.

W e denote by the family of all the u n ital product subsets of X. We introduce two kinds of equivalence relations in a s fo llo w s. =iliENEi and E '=IliE N E 'i be two elements in a

D efinition 1.1. E is co fin a l with E ' if the condition

(CF)

E,Eivii(ECLE)17A4. Using this fibre bundle, I. M. Gelfand and others have constructed irreducible unitary representations ( = IU R s ) of the group G in [17]. Let us begin with the following important fact. - -

Lemma 1.6. The subset J? of X belongs to 91tx. For any u n ital product subset E , the Product m easure vE is carried by g in the sense that, f o r any L E fft( E ) , L n k is also in 9J1(E) and (1.16)

I■E(L)= vE(Ln)?) .

P ro o f T ake a system of countable o p en b ase U.,}3EN of M such that each U., is relatively compact, that is, the closure cl( U ) is com pact. Then .g is expressed as 3

n

3EN

u {,T=cx,) xl

u.,}1, we still choose open paths Pi connecting E r with for 1 j< i so that - i

;

(1.24)

c1(P ,J)n c1(E k")= 0 f o r k * j ,

,

and that the union of E i " , 1 j i , and P i k ,l k < j i , does not cover any () fo r 1> i (if necessary, shrink Ei ° a little). STEP 4. Now assume that El", E2", •••, E," have already been chosen. To choose Er-1, we start with

(1.25)

B'q+1=E

\til),,C1(E.inUi,k .).,,c1(Pik )}

839

Irreducible unitary representations

and shrink it if necssary to get an open set B

g + i

c B 4 1 such that

p (B ± 1 \ 1 3 ,1 )< 2 -(" " , cl(B g+ i)n cl(E , (°) ) = 0 (15 j

(1.26)

Then we follow Steps 1 to 3 for i = q + 1. Thus we get j < i . The induction process is now completed. Let us evaluate the sum EzENP(ETeE,m). Note that (.)

E;'0E,mc (E;'CDR) U ( B ; c ) a ) u m e E

with get

and take into account

R ea=

q) .

and .13 „ 1 -5 0

)

R \ B and (1.23), (1.26), then we

E N ti(E 'O E i ° ) 5 EiEN,a(E7C)Ba+2 . (

)

On the other hand,

n( U ci(Ei(o))u ,, y.

E7e.B ;=E7 \

k

< i

ci(Pio)

i

and so fi( E 7 e .B ) 5 E ,a((j < E E7)(1c1(E; °))+ EP(C1(Pik)) i< c. kQA, for (H, V(H)), we extend every çoE V E to a function f = Q 1 1 9 on X as 2

I ,

(2.8)

Q 17ç9=

E 6 .e.(R (a)• 11( 0 9

or m ore exactly, for x E E ' a n d cs a.,, (Q119)(xa)= 17(6) - 1 (9(x)) ,

(2.8')

recalling that E' a (c E a .) are mutually d is jo in t. Then f =QH9 satisfies

f (x a)= M a r f ( x ) ( x E X , a E a .)

(2.9)

and f ( x ) =0 outside llae s ,.,E 'c . Put

Ilf11 =f,11f(x) KundvE(x), 2

then we get a H ilbert space SCrE, , isomorphic to V E through the map Q . T he H ilbert space 3C(E) for —(17; i , E ) is th e o n e generated by the family ATE', E ' E : I ,

(2.10)

M

( I ) = V E 'e ( E )

ATE' .

Here, for two elements fiESCrEw with E m E , their inner product is defined as follows: take an F E W (E ) such that F a ( a e a s ) a re mutually disjoint and fi(x )= 0 or f2(x)=0 outside L10Es_F(5, then -

(2.11)

< fi, f 2›.41 (1) .

—

f V ( I1 ) d 2 ) E ( X ) .

We see easily that the map Q/7 gives a n isomorphism of H ilbert spaces ,(/ ') and SC(E). Remark 2 .1 . The H ilbert spaces ,(2') and 3C (/) are actually generated respectively by and 3CrE, = Q u a with E ' E (strongly cofinal):

(2.12)

,(2 ')= V

, M (E )= EfV,E I C [.

Therefore th e subset A o f step functions o f th e form ço=x.30v with BE 9JZ(E)1E', for some a n d yE V (H), is to ta l in ,f)(E), and so the subset Qi7A={Q/79; ipE A } is to ta l in J C (/ ). In particular, the H ilbert spaces VX) and JC (I) are separable.

2 .3 . Unitary representation Tx o f G . A representation o f G is given on S C(I) as (2.13)

T . ( g) f (

x )= ,0E(g- '1x)"2.f ( g - l x )

842

Takeshi Hirai

for fE 3C (2'), gE G and xE X , which we denote by the same symbol TE with that on VE). The unitarity of 71 is clear if we note that the domain of integration F E (E ) in (2.11) can be replaced by g F if fi and f2EJC (Z) is replaced respectively by TE(g)fi and TE(g)i2. For the continuity of G D g l-- TE(g), we should first mention the topology of G . The group G =Diff o(M) is equipped with the usual D-type topology or the inductive topology according to the family DiffK(M), K compacts in M, where DiffK(M) denotes the subgroup of G consisting of g with supp(g)ŒK, and is equipped with the topology of uniform convergence of every derivative. Then we have -

Proposition 2.1. T he form ula (2.13) gives a continuous unitaly representation o f G = D iffo (M ) on the Hilbert space C (f). 1

Pro o f . Since the unitarity is already known, it is enough to prove the continuity of GDg1-3 for fi, f2 in a fixed total subset of M(X). Therefore, by Remark 2.1, we can take as A, f2 two step functions of the form E and y e V(//). fi= Q uso, w ith ço,=xB , Ov.,, B J E9N(E)IE ' for unital Then ,

_ v ( H ) .f E ,2 , p E (g -li x )1/2x B , , , (g - i x )' pE

= v(in• f

n 13,2,

x 13,

2,(X )C /V E (X )

( g il x )112 dv E ( x )

Further we can take as B ' a subset of the product form f3 =11,ENB, such that B » = E , for i > 0 . Then, by (1.12) and an additional discussion on the interchange of "infinite product" and "integration", we get (

(3 )

)

(3 )

)

< T E (g)fl, f2>

—

< V I , V2> VC/7)

.

H

e

ie N f g B n)r)Bi (2 '

PM (g - 1 , X i ) 1 1 2 C / ( X i )

.

Now, fix a compact K M and take g E D iffK (M ). Then, by assumption on the measure ji, there exist positive constants ci(g), c2(g) such that m(g ; P) 0 and c2(g):- c2< co for any g in a certain neighbourhood o f e=identity in D iffK (M ). Thus an evaluation similar to that in § 1.4 proves that, when g tends to e=identity, the infinite product in the above expression of < T1(g)fi, f2> tends to V (II)

•

n /31(2) ) =

H 1. ( B EN 1

oh, v2> v(n)- vE(B(i)n B(2)) =.

The continuity of g■- T i(g ) is now completely proved. ,

Q.E.D.

843

Irreducible unitary representations

Thus, for a d a tu m =(/7 ; p , E ) with a UR 17 of ao and a unital product subset E of X with respect to p , we get a UR Tx of G . For another datum X' =( / / '; p , E ') but with the same p , we have Tz = T E if 11' 1 1 (unitary equivalent) and E' — E (cofinal). , -

Remark 2 . 2 . To study irreducibility and equivalence relations for TE's, it is often convenient for us to use the realization of T x on V E ) =V E ,E M ,

,

,

and utilize the family of subspaces for explicit calculations. These subspaces play a role, for the Hilbert space , (2'), something like local charts for a manifold. Symbolically speaking, global structure of the manifold M is reflected in the structure of V I ) or JC (E )=Q H (X ) at the point how to patch together these subspaces a or ,gCrE. = Q/7,M , whereas in each VE only local structure can be reflected. Anyhow we have enough reason to keep two kinds of realizations of TE: the one on V E ) and the other on ,4C(Z)= Q .4 (Z ). ,

,

Some fundamental lemmas

§ 3.

Here we collect some lemmas which will be needed

later.

3 . 1 . Elementary representations o f diffeomorphism g ro u p s. Let E1, E r be mutually disjoint open subsets of M , and put

E 2 , ••",

‘. 1 = 0 L 2 (E ) w it h L 2 (E i) =L 2 (E1;

G1 = H G 1E, w i t h G IE ,={ g 16ir

,

G; supp(g)cEi} .

Then we have a natural representation of G1 on that on L (E1) of GE,: forg = ( g ) GI and y = (Y i)E lli6 ii-E i

as the tensor product of

2

i(g ) f)(y )

(3.1)

H

p m

(g i -i ; 3 ,,)112. f (g

-1y

)

the Jacobian om(g; p) for p E M in (1.11), and gy =(giy i). Note that G E = G ( E i ) D i f f o ( E i ) . Let the connected components of Ei be E , JEJ (we admit the case ILI = o e ) . Then GlEi.,= G(E1) for every E0, and G(E1) is equal to the restricted direct product of G(E1.)'s, and is contained in the direct product of G(Ei )'s : with

t

1 1

0

I

f

(3.2)

is) = We get easily

fiG ( E .

the following

Lemma 3.1. T he representation L 1 o f G1 o n i i s a direct sum o f IUR s which are not m utually equivalent. P ro o f . The group G1E1 is equal to M e.,,G (E k i ) and the natural representa-

Takeshi Hirai

844

of each G(E u ) on L (E 1) is irreducible, and the decomposition L (E1)= V E ,L (E u ) for each 1 I r gives, through tensor product, a desired direct sum decomposition. Q . E . D . tion

J

2

2

2

be also mutually disjoint open subsets of M, and put

Now let F1, F2, .N = 0 L 2 ( Fi) ,

G 2=

H G(Fi),

W 6S

then we have a natural reprensentation L2 of G2 on ,N similarly as (3.1). Let us compare two representations L i of Gi (i=1, 2). Put =Ei n F ; for 1 i B 1 .=E 1 \c 1 ( U F; ) , B .0 = F i \c 1 ( U B a ) , i

and L0= {1, 2, • , (3.3)

ti(Ei

Then L

2

J

s

r , 001, Lo={1, 2, •, s, c o } .

,V=-EED 0 1 i r J

where j= (ji,j2 ,• • • , G12

H

=

p ( F ;\ i eUl . B ,)=0 .

U B ii)=0 , JE J.

and therefore

( E 1 )= V E 1 .L 2 ( B 0 ),

(3.4)

Assume that

,N"="E(9 0 LA Bii i ),

i

jr ), jiE J .,

and

i2, •--, is), ii E lo o .

We put further

G(B u ) c ci n G2 ,

where B ...=0 and G (B 1 )={ e ) for B11 = 0 . Since F i=0 for any j , we may understand that each G ( B 1 ) acts trivially on 1 ' 2. Each G (B u) acts naturally on L (B 1) and trivially on L (B1y ) if (i', j') * ( i, j) . For j=(ji, j2, •••, j r ) and i2, •—, is) in (3.4), we put 2

,f)(j) =

2

L 2 ( B 1 ji)

,

,

f)(i)

=

C ) L

2

( B1 5 j )

r } and [i]={ (i i , j); 1 j Further consider the set of pairs [j]={ (i, ji); 1 ,s}. In case [ i ] D [ j ] (and necessarily r -5s), we define ,( [i ]\ [j ]) as the tensor product of LA B, j ; ) over j such that (i , j) E [ i ] \ [ j ] . Now suppose that [i] and [i]\ {j]c {(0 0 , j);1 s} , then changing the order of the factors in the tensor product, we have a natural isomorphism 1

(3.5)

0,,: ,f)(i)—

(i)O V [i]\ [j]),

On the other hand, for any OE,f)([i]\[j1), MOM=1, we have an isomorphism, under of , ( j ) into ,( j ) 0 ,V [i ]\ [j ]) given as /di00:ço—> ço0 0 , where I d ; is the identity operator on , ( j ) . Then 0 i , -( M ;0 0 ) gives an isomorphism of ,f ,(j) into V i ) . - 1

845

Irreducible unitary representations

Lemma 3 . 2 . Fo r two representations (Li, t ) o f G, f o r i =1, 2, let A : be an intertw ining operator o f L11G12 w ith L21G12. A ssum e th at the condition (3.3) holds. T h e n A is a sum o f scalar m ultiples of the natural isomorphisms ,

0(Ic JO

(3.6)

,

where O E V [i]\[j]) , and the Pairs ( i, j) satisfies the condition [i]D [j] , [i]\[j]c { (c 0 , j);1 j s } . .

A proof can be given by using Lemma 3.1. In summary, we can say that dividing E i's and F i s into B =E 1 C 1 F, and their outsides B ,- , B .,, it is enough, for intertwining operators, to pick up in (3.4) the factors of a n d having the same B u 's in common. 3 . 2 . Infinite tensor products of representations. Similarly as above, we get the following, for an infinite tensor product. Let E' =IL EN E; be a unital product subset i n ( E ) such that each E ; is open and connected. Put G (E )- --H ;ENG(E), the restricted direct product of G (E )=- G1E/. Then G (E ') is contained in G naturally. We define a natural representation o f G (E ') on IE. = L ( E', O E M ', vElEy) as - --

,

(3.7)

LE,(9)f (x)=PE(g

- l

ix )" f(g 2

- l

with pE(glx) in (1.14), for g G(E'), f Lemma 3.3.

,

-

2

x) ,

f)1E.

,

and x E E '. Then

The unitary representation

LE'

o f G (E ') on

is irreduc-

ible.

Proof. P u t = L ( E , die I E ) , so = xEd

an d 9 = ( 9 ,) iE N . Then f)1E is canonically isomorphic to the tensor product Ofeiv i=®,eN çoj of with respect to the reference vector ço. Because of (1.14), the representa2

,

,

,

tion L E ' of G (E ')= T L E N G (E ) is equivalent, under the above isomorphism, to the outer tensor product OfENL, of reprensentation L , of G (E ) on where (3.8)

1—(g)sb(p)= p m (g '; p )i' 0 ( g 'p ) -

2

-

for g e G (E ;), O G ,, and P E E . Each L , is irreducible since E ; is connected. Therefore its tensor product is also irreducible, whence so is LE '. Q.E.D. 3 . 3 . Stru ctu re of a subgroup G ( ( E ') ) . Assume now n = d im M 2 . For a subset E '=1 1 i.N E ;, E C M , of X , let G ((E ')) be a subgroup of G given by

(3.9)

G ((E"))=-{ g G; g E = E r(i) (iE N ), 3 crE - } .

846

Takeshi Hirai

Let E' =ILENE; b e a (UPS3)

—

unital product subset i n (E‘) satisfying the conditions (UPS4) (replacing Eim there by E ) , which exists by Proposition 1.8.

Let us study the structure of G((E')) for such an E'. By (UPS4), for any i * j , c l(E ), c l(E ) and Cz,=c1(Uk*,,,E'k) are mutually disjoint, and there exists an open path Pi, connecting .E;, E; such that cl(P1) n c „ , = o . Put M = E U E U P1 . Then Mi., is a connected open submanifold of M and s o G(M,,)=Diffo(M,,) is canonically im bedded into G=Diffo(M). W e shall construct an element g ,, e G(M1 ) such that 1

1

1

gi,E;=E; , g i , E ; = E ; .

(3.10)

Then g1j1E'k is the identity map on .E;, for k* 1, j.

Thus we get the following

Lemma 3 .4 . L e t n=dim M 2 and E '=1 1 i.N E b e a u n ital p rod u ct subset i n (E ) satisfying (U PS 3 )— (U PS 4 ). Then there exists fo r a n y d E a o , an elem en t go-EG su ch that gcrE = Ea (I) and godE;= id en tity on E i f 6(0= i. P r o o f It is enough to construct for each 1*] an element gi E G(M i )c G in (3.10), and put gc,„=gi, for the transposition au (i, j ) . STEP 1. Inside of P , we can choose two non-intersecting simple paths C1 and C2 connecting a point x iE E ; w ith an x EEJ. i

f

—

0

i

Figure 3.1.

We give to the path C1 (resp. C2) a direction from xi to x , (resp. x , to x i) . By (UPS4), E ; and E ; are respectively diffeomorphic to open balls B io c E with center x , and B o cE .; with center x , by elements h i, h,E G with supports in small neighbourhoods of cl( E ), c l(E ) respectively. N ow choose a series of points x,i, x 12, •••, xir on CI (resp. Xi i, x,2, •-•, x,,s on C2) and sm all open balls B 11, B12, • • • , B 1 (resp. B 3 1, B1 2, •••, B 18) with center Xii, x 12, •••, x ,r (resp. x,i, x,2, •••, x, ) respectively in such a way that

ci(Bih) n ci(B t)= 0 for any k, i

B10 n B i, *0, /3 1 1n Biz*O,

Bio n B i s * 0 STEP 2.

Put

,

Bir n B 0* 0 ; 1

B 1 in B 1 0 * 0

.

Irreducible unitary representations

847

Figure 3.2.

, , y < s 13.51) , = E\

r

u

sB B U 1 : E ' E ° . F urther, °> as seen below, we can chose E so as to satisfy one more condition (UPS5). °> For any so chosen E , the Hilbert space X ) = % ( ^ ° ) is spanned by C " s with E ' E ° which satisfy (UPS3)—(UPS4) too. For a un ital product subset F = H N F i satisfying (UPS3)—(UPS4), we consider the following condition. (UPS5) For every N>O, the complement M \ cl(LJl>NFI) of cl(Ul>NFI) is connected. L em m a 4.2. Let F = fll N F l be a unital product subset satisfying (UPS3) — (U PS 4). T hen, cutting o f f a sm all part of each F1, iE IV w e g et a u n ital p ro d u c t su b se t F'=H ,N F', F,C C F1 , c o f i n a l to F, w h ic h satisf ie s (U P S 5 ) together with (UPS3)— (UPS4).

Put A =M \ cl(LJIENFI), then A is not empty. In fact, if A 0 , then M = cl(1J N F I)= cl(F l)U cl(U 12F ) on the one hand, and cl(F i) and c1(U12F1) are mutually disjoint by (UPS3) on the other hand. This contradicts the connectedness of M . Since A is open, its connected components A1, A2, a re a t m o st countably infinite. We connect A 1 with A 1 + 1inductively as follows, getting F , from F accordingly. First connect A 1 with A 2 by a (rectifiable simple) curve C . In case C meets with F1, we shift continuously the part C fl F of the curve very near to the boundary c l ( F 1 ) \ F 1 , and pare off a small part of F1 together with the shifted curve inside F1, thus getting a connected open F » C F 1 such that 1i(F \ F 1 ')< 3 1 'i(F1). S o th a t F ( l > = [ J i E N F i ( i s cofinal to F , and A 1 and A 2 is connected outside cl(U1ENF1' >) by the shifted curve. Secondly, to connect A2 with A3 properly, we work similarly with a curve C connecting them and with F= H F»>. P arin g a sm all p art o f each F ' if necessary, we get a connected open F 2 C F » su ch th at 1 i ( F » \ F 1 2 ) < 3 2 u i ( F 1 ) an d th at a continuously shifted version of C is outside cl(U1ENF1 2>). Now assume that for each iE N , we have chosen opens F 1(l>D F 1(2 * F 1 ' su ch th at i ( F » \ F » ) N F) for N 0 . L et u s prove that BN a re connected for N 1 , b y induction on N . N ote that, by (U PS3), cl(Uz>NF;) is a disjoint union of cl(Fk+i) and c l(U > N + IF ). Then we see BN+1=BNUcl(Fk+i) for N 0 . A t first, a s seen above, Bo is connected. Since cl(Fk+ i) is connected, we h a v e a connected open neighbourhood F " o f it, d isjo in t with cl( U > N + 1F ). Then BN+1 = B N U F " , B N n F " * O , a n d therefore BN+1 is connected. This completes th e in d u c tio n . So the condition (U PS5) is satisfied. Q.E.D. -

-

)

)

(k )

-

)

4 . 2 . A lemma fo r irreducibility. A ssum e that E ' — E is taken to satisfy (UP53)— (UPS4) a n d (U PS5), and that E';--,E ' a r e taken to satisfy (U P S3)— (U P S4). Th en w e see, by Lem m a 3 .5 , th e r e s tr ic tio n TE-(g)— TE,0,(g)13CrE f o r g E G ( ( E ') ) gives a n IU R o f G ( ( E ') ) . T o o b ta in the irreducibility o f Tz(o) o f G from that of the family ( TE , A l ), we apply an elementary lemma given below. (

-

(

)

)

,

,

,

Lemma 4 . 3 . L e t H be a group a n d T its unitary representation o n a

Hilbert space A ssum e there exist a fam ily of subgroups { H8}8.4 an d that of subspaces { s}se.d such that (a) ,f)8 is Hs-invariant an d Hs-irreducible; (b) is spanned by the fam ily Wsls.A; there exists a finite sequence 81=8, 8 2 , • • • , 8 ,=8 ' such (c) f o r any 8, that s, n , , , * ( o ) f o r l i < r; (d) f o r som e 8 E , th e IU R o f I I 8 . on ,f) 6 0 does not appear i n t h e orthogonal complement ( )-Lc . t Then the representation ( T , V o f H is irreducible. ,

0

0

either P ro o f By th e assumption (d), any H -in v arian t subspace of contains th e H8 -irreducible subspace ,18, o r is orthogonal to it. Therefore Now take any aE J. containig there exists an H-irreducible subspace f o r 0= < i= (0 ) < s with 8,+1= 6. We can find a', •••, as such that ,f)8, n ,8 , , Since c o n ta in s S58 n ,* (0 ) and 3 i is H -irre d u c ib le , we see that d o e s contain the whole . . Similar arguments prove inductively that c o n ta in s ) a n d 1 5 8 - ,= 8 . Thus contains all of .f)8, (3E J . This proves that "f 82, •• • 3

0

0

0

r

851

Irred u cib le u n ita ry rep resen ta tion s n '= t by (b).

Q.E.D.

,

4 .3 . Proof of Theorem 4.1. We prove the theorem by applying Lemma 4.3 for H = G, ( T , ,0 = ( T ,0,, JC(1")) and the set of parameters E

( )

J = I E '; E ' E ' , with (UPS3)—(UPS4)} . ,

For 8 = E 'L J we take Ha= W E )), )8 = JC(k . Then (a) is proved by Lemma 3.5, and also (b) holds. So we prove (c) and (d) here. First consider (c). Take E', E" E LI a n d p u t i =,gCrE , ,f)3=‘.4CrE . Let us find E EJ such that ,N= ATE. satisfies ( 4)2*(0), ( 0 ) . S i n c e E', (°) E " are strongly cofinal with Em, there exists an N >0 such that E = = E , for i> N . We see easily that there exist a o E N and an NI, O N1 N, such and that { E U E mutually that E n Ez,.(,)*0 for N i< , ;1 a r e ) (3) d is jo in t. P u t E " = E' , E = E" a and ,

,

(2 )

,,

-

(

E(2) = ( I I F) X E M w i t h I

V )v= H E1( ,

N

N

)

13

)

then E and with Fi= E iU E i) for 1 i NI and F i= E,;(,) for N I< E J . P u t ■N= 3CrE(2), a n d n o te that ..SI=ATE = M fiii) a n d .N=A7E--=-ATE,3,. Then we hav e ,in ,i+ ,*(o ) for 1 i2 b e c a u s e 1 E (E (1E + ) * O . This proves (c) with r=3. Let us now prove (d). Fix a 8 0 = E '. Take an arbitrary 8=E" from and study the decomposition of the subspace along the orthogonal decomE (», there position f) = 600 ( s0) . S in c e E ' and E " are strongly cofinal with exists an N > 0 such that (2 )

(

,

(i)

,

,

H

1)

±

,

E '= (

(i

A i)x

,E " = ( H

Bi)X Enr

where Ai's (resp. Bis) are mutually disjoint open subsets of M \ cl(U i> i

(°)

).

Put

A in B C i o = A i\ cl(U i N_13;) , Coi=131\c1(U1,NA1) ;

then A i \ Uoi‘NCi.; a n d B A Uo,i,NC i a r e o f measure z e r o . Therefore, modulo null sets, B =II w,N.B.; is a disjoint union of subsets of the form D = with D = C fo r some if . Let DB be the set of all non-empty such D c B a n d p u t [D] =D x E M , then AT ,=7.9 , Similar statement is also true for A=111,i,piAi so that T a k e a D DB. T h e n , each component Di is either contained in some A i o r d isjo in t with all A . Therefore we have two cases: (i) AWL)] ciqi in case AYE A for some csE N , , (j* and (ii) M r D .1—gCri in case some D is disjoint with all A i or two Di , j') are contained in the same A i . Thus we get an orthogonal decomposition of f o r each 8 J, as ;

0

)

E

,

[

]

i

Takeshi Hirai

852

,f

,f ,7,. 7

. 2j77.

( 571 )

, 7,_,7g.cr-E,, _,75,1e),7f,,,2

where

and ,t•8 are respectively the direct sums of A' ffp, in the cases (i) and

c

..f) 8

7,0

2

(ii).

Now consider a subgroup H808 of 1160=G ((E)) defined as Hs o a— V DEDAGD

w it h

GD

—

—

V15i5NGIDI =

The representation of GD on ,g1ftD1 is a multiple of the natural IUR on / 2 (D ), and the one of GD on other JCifD ) with D 'E D A U D B , D '*D , does not contain the IUR on L ( D ) (cf. Lemma 3.1). Therefore we see that the representation of the subgroup 11 8 0 of HS() on the space , 8 = ,g CfE is disjoint with that on .V c ( . 8 ) L. On the other hand, the orthogonal complement (f)8 °) is spanned by , 8 E 4 , because the whole space 3 is spanned by ,f)8's and each h a s the decomposition (5.1). Therefore the IUR of 1180 on f h o does not appear in ( 80) =V8E 4 . This is exactly the assumption (d). Thus the proof of the irreducibility o f T x is now complete, and so Theorem 4.1 is proved. ,

2

0

-

7 ,

-1-

-

0

2

-L

2

§ 5. Equivalence relations among the IURs

Let Hi, 112 be two IURs of a., and E = I I ,N E ,, F = 1 1 ,N F , be two unital product subsets of X with respect to a measure # on M . Put L'1=(//1; p, E), E 2=(112; p, F). We study here a criterion for the equivalence Tz,"" 7 12. -

be the group of all permuThen it acts on X from the right: x a=(x a(0 ).N for aE g. and and accordingly on Hi as . Further ge. acts on

5 . 1 . Natural equivalence relations. Let

tations of N

.

x = (x ,),E N E X

aa=ac a ( alli) ( 6 ) =1 1 1 ( a a) =1 7 1 ( a i (la) ( a c e . ) . -

- 1

By the action x - - x a on X , a unital product subset E = E a=lliE N E a(i). Further, if E ' E , then E' a— Ea and v Ea(E'a)=Ili.N p(E'a(0)=11JEN p(E;)=

iE A r E i

is sent to

.

This means that the action x — x a on X gives an isomorphism of measure spaces (X , 937(E), 2)E) and (X, TI(Ea), vEa). the Since E is cofinal with Ea if and only if a belongs to the subgroup measure spaces coincides with each other if and only if a E a o . Put, for f = (H ; p, E ) and aEgoo, e c o ,

(5.1)

=(aH ; p,

and, for f E 3C (E ), (5.2)

(R af ) ( x ) = f ( x a) ( x E

.

853

Irreducible unitary representations

Then, as is easily seen, R a g iv e s an isomorphism of Further we obtain the following

JC ( E )

onto M(aE).

Lemma 5 . 1 . For a E , th e map Ra gives an isomorphism of IURs (TE, i ( ' ) ) and ( T az ,M (aZ )) o f G.

5 . 2 . Equivalence criterion. As a necessary and sufficient condition for unitary equivalence T11:=' T 12, we will get the following rather simple criterion.

Theorem 5 . 2 . A ssume dim M . 2 . L et L'i =(111; p, E) and 12--(172; p, F) be as above two parameters of IUR s of G =D if f o (M ). Then, ( 71 , 1 C C )), i= 1, 2, are mutually equivalent if an d only if , f or some element 1

112 a n d E— Fa

(5.3)

(cofinal) .

-1

The rest of this section is devoted to prove this theorem. 5 . 3 . Relation between E and F . T o prove the above criterion, let us begin with the relation between E and F.

Proposition 5 .3 . A ssume TE1= T12, then necessarily E— Fb f o r some bE -'

The proof of this proposition is rather long and is divided into several steps. STEP 1. As is seen in § 1.8 and in § 4.1, we may assume from the beginning that each E and F satisfy the conditions (UPS3)—(UPS4) and (UPS5). Recall that (5.4)

(1 0 = ,,\( ATE'fi E

, (E2)= ,y,,,gq2.,

,

(

F

where we can also take E ' and P " satisfying (UPS3)—(UPS4) too. Let A : JC(E1)-gC(E2), be a non-zero intertwining operator o f T z , with 71 . Since both T z , (i=1, 2) are irreducible, we may assume A is unitary. Denote by P l ci■the orthogonal projection of JC (E i) onto JCT., and put A F , i,E . Then A p n E w , viewed as a map (

)

2

i

(5.5)

A F(')E in:

intertwines

'MT"' =

T z l G( 1 ) i

Qm I E

1) -

-

A T F2' 1) =

6

2170 F

2(1

'

w ith TizI Gm for a subgroup G w = G ( ( E " )) ) n G ((F "))) of

G.

By definition, for gE

there exist c and r E a . such that

gEi =E M ( i E N ) , g F (1 )

)

P u t hul(E ") =c1(U i.A rEi m (

)

;

(1 )

and

=E -

1)

(;)

( jE N ) .

Takeshi Hirai

854

n

= F JP = E i")

(5.6)

,

n , -- - E iw \hul(F" ) ) ,

and

t h e n hu1(E ) (1 )

hul(F

(1 )

Fj1 = F 3 " \ hul(E ") , )

(

)

are invariant under g and

)

g n )= E M ), (; ), g E N = E W i). g n o ) =FW.i)- •

(5.7)

Put further Ei(2)=

(5.8)

F . 2> = E F w (disjoint unions) . EP ' " , F1 = ç

1

J

jo o

T hen E, ==-E an d F, ( ' ) F . » ) modulo n u ll sets, and therefore, for E>2 = and F =ILE N Fi , H ,E N E (1 )

2)

(2 )

(2 )

r

i

E cl)

—

(2 )

eX T 1 2 1

,q ( rF 211)

—

17 2 F 12) c..41 1 .

Let Ck , kE K , be all the connected components of .E9 =FP , and F J . T h e n G ' contains the restricted direct product G of G c,,=G(Ck): G =11'EAT G(C k)c G ' . To study the action of G , it is convenient for us to ,212), _ ,2.( 2 ,= 0 h riThe Hilbert space ,•, E ,2,= use V 2(2 '= -IE.(2 0 V(T11) instead of Mi L ( E , V(E )1E , dv ElE ) is isomorphic to the infinite product OrENL ( E, ) of STEP 2.

)

(

)

(

(2 )

7

2

(2 )

,

)

(2 )

i

(2 )

l

,

-

)

2

(2 )

(2 )

L 2 (E i(2 ))= E

l9)

W600

L 2 (ET )) =

E

el

kEKli

L 2( C

(2 )

k)

w ith respect t o th e re fe re n c e v e c to r 0 = ( 0 i ) i . N , O1=xE, ./11XE, 211, where L (E1 ) = L (E im , cliflEi ( 2 ) ) etc. and K i i = { k E K ; CkE Ei ( 2 ) }. F ix an N >0, ,

2

, ,

2

(2 )

and put ,N k ) =

(5.9)

1

N

L (C i)

fo r

k=(ki, k2, •••, k N ) E

H

15 i 5 N

K li

then we have a natural isomorphism

(5.10 )

•f)1E(2) -='(Ek('I■i( k )) 0

(

„

0 ; ' L 2 (E i( 2 ) )

,

w h e re 0 '=( 0 2 ) ,>N . B y m eans of th is, a c o n sid e ra tio n sim ilar to th a t for Lemma 3.2 proves that A F ,1 ) E i n = A F ,2 ) E ( 2 ) kills all the components f)1(k)0(0r>N L ( E1 )) w ith k —(ki, k2, • • , kN) for which k‘=00 or Cki c.E2,2=E, \hul(F" ) for some i N. T h u s Afi,2)E(2 k ills all the components containing a factor from L 2 (n 2 ) for some i E N . (To see this, we actually compare the action of GiEi,pc G .) From this fact, we get the following ,

2

(1 )

(2 )

5 .4 . A ssum e that there exists a non-z ero intertwining operator A

Lemma o f TE, w ith (5.11)

)

TZ2.

Then,

EiENti( n . ),) < 00 ,

EJENp(Fil) ) < 00 ,

Irreducible unitary representations

855

f or any E m E and Fu F , where n r= E ,m \h u l( F m ) , F g \h u l( E m ) as in (5.6). )

P ro o f Put .M.;-) =Ei&■TE.1 ) , then Ei =

+ E22

(2 )

and so

L 2 (E i ( 2 ) ) = L 2 ( E 2 ) 0 L 2 (E 2 2 ).

Since A ,2, ,2, kills all the components containing L ( E ) -factor for some i E N , we get A F2E (2)(x E -)=0 for any E " E (strongly cofinal) if Ei.N p(Eg1) =0 0 or equivalently if II i» o p (E n =O . This means that A F ( 1 E 1 =A F 2 E 2 =0 . On the other hand, if E iEN ii(Eg2)=09 for some pair E m E , F'>— F, then it holds also for any pair E '— E , F'>— F. Hence we get A =0 , a contradiction. Q.E.D. A similar argument for 11 proves the assertion for F.P.2's. F

E

(2 )

-1

STEP 3. Now replace Ei and E respectively by E i = E i + M = F + E5!2. Then, for E =11iENEi and F =H ENF , we have m

(1 )

;

(1 )

(3 )

(3 )

(3 )

(3 )

; F(3)DF(i)

E(3)DE(i)

hul(E ) = h u l(F (3 )

(3 )

(1 )

i

;

;

, FI 3)

(3 )

F(3),,,, F0)

) e=hul(E ) U hul(Fm)) . (1 )

Note that

then we obtain

Lemma 5.5. A ssume that there exists a non-zero intertwining operator A o f TE, w ith TE . Then A is approximated strongly by the family A F(i)E.= oA .P lin with E m E , F F such that E, an d F,,a are open and hul(E a ) 2

(1 )

(1 )

)

)

=hul(Fm ).

Corollary 5.6. A ssume TE, --- T12. Then, replacing unital product subsets E in Z1 and F in E2 by their cof inal ones, we can assume that h u l(E )=h u l(F). (Here the conditions (UPS 3)— (U PS 4) are not necessarily satisfied.) -

5 .4 . From hul(E)=hul(F) to E F b ( ] We continue to study the relation between E and F , and wish to get E — Fb for some STEP 4. By Lemma 5.5, we may pursue A F,i)E,,) with hul(E ) — hul(F" ). In this case, E22=0, F1, 0 =0 and (1 )

)

1)

(5.12)E 1 2 = = E

je N

= E

k e K li

Ch

E Ck, F; = ieN E F J P =k eK 2j (2 )

p ( E i n E i ( 2 ) ) =0 ;

P (F j

")

\ F j (2) )

=

0,

where K zi= {kE K ; C h cF ; }. Consider the action of C =I L E K G ( C k ) on both of ,VE',i) ,f) E, 43) V ( / / 1 ) and 0-2.) =‘,F( )® V(/72). Note t h a t G a c ts triv ia lly on the second factors V ( /7 1 ) , i =1, 2, a n d t h a t n-,1E(i — E 2 i s decomposed as in (5.10), and — , f ) I F ( 2 ) as in 2)

(1 )

---

1

-

(2 )

,

,

,

,

Takeshi Hirai

856

(5.13)

=(

N(12'))0 (A,1- ( F1 )) 2

/

2)

for N ' > 0 (the reference vector is omitted for 0.,>N (5.14)

f)2 (k )=

L 2 ( C k ,)

fo r

),

,

with

VAT ) E 1H N K2J.

k '--(16, /6,

,

,

Considerations similar to that for Lemma 3.2, on the action of G , give us the following crucial lemma. (2 )

T he im age of A p i,E (i)=A F.E 2 is contained in the sum of

Lemma 5.7. the components

i (2 ))) )2(12 ) 0 ( A , L2 (F

(5.15)

,

f o r which the param eter k '=( k i, k , , k 'N )E rl1 s ,N K2., satisfies the condition ,

,

k'N,, no two o f them belong to the same K1, i E N .

(NT) among 16,

STEP 5. Denote by PAP the orthogonal projection of F2 onto the subspace spanned by such subspaces in (5.15) that the condition (NT) holds for b or them. Then it is enough for us to prove PN ->0 strongly in case E F In fact, if so, w e have Ap1,E(i)=AF(2,E2,=0. This E / F b for a n y bE means A = 0 by Lemma 5.5, a contradiction. To prove PN - ) 0 is reduced to a problem on series of real numbers as follows, by considering PN (x.p) 0 (N '-* co) for any u n ita l product subsets F ' E F which are strongly co fin al to P >. P u t c,,, =p ( E g ) =p(F1, ) , and we consider n = P r grouping C k's, instead of Ck them selves. Put l

(2 )

,

(2 )

,

--,

,

2

(2 )

)

1)

)

d1= (E1> >)= 2

f

NC1

P(Fi(2 ))=E i.N c ii

,

and we come to the following problem (N ' is

then p(E1 e F ) =d 1 + replaced by N ). (2 )

2)

Problem 5 . 8 . L et cii M f o r i , j E N . Put d i=E ;E N c u , e i=E i& v c ii, and assume that d 1 > 0 , e >0 ;

i.N d i,

j

N );

IL E N e; are (unconditionally) convergent.

For N >0 , expand the product II (re s P .ili,,s N d z ) in terms o f c i,'s and sum of monomial terms c c,22---c„„N (resP. c C2jz civ.m) let p N (resp. qN ) be the such that (DE) i 1 2, ••-, iN (resP. ji, jN ) are different from each other. 1

,

A ssume further that

1

Irreducible unitary representations E (di+ eb i -2 ci,b

(5.16)

(

ieN

Then, does there hold

Note that (5.16')

)

pN—> 0

)

f o r any b E & .

) =00

o r qN-■O as N->00?

is expressed as

(5.16)

E c0=c0

i,ieN i* b ( i)

5 .5 .

(

857

for any

Solution of Problem

bE & .

5 .8

(Step

1).

F o r a finite subset

J

of

N,

consider the product H je J e j

and expand it in a (possibly infinite) sum of terms (5.17)

H jc jj

w ith

( 2 ,) ,E J E N

J

Denote by p the sum of all such terms (5.17) that the sets of indices ti,; jE consist of different integers. Then PN = A I N for IN = {1, 2, 3, •••, NI, and we get easily the following j

Lemma 5.9. Fo r two disjoint finite subsets J ,

J'

of IV,

PI UP PJ • PP •

To treat pN or p , we can normalize (c u ) as follow s. Put 6 = c d e ; with ei =EiENc u , then E ie N C ;;= 1 . Let p b e the sum for (6 ) corresponding to p for (c ). Then, p;=p j /(ILE j e; ), and f I J E J e J is bounded from above and below for any finite J c N . Therefore the assertion on PN for (c ) is equivalent to that for ( 6 ) . Hence, to treat p N , w e m ay put the following additional condition on j

j

0

u

(5.18)

(j E N ) ,

e., ---E N c „ , = 1

and in this case w e m ay call (c A ,,e N a stochastic matrix of infinite size. Here we should note that 11,Nci[, with d,' = E ENc;,, is again convergent. This can be seen from that every term c •••c 'N ,N in the expansion of and that all the products ili,,,N X is a multiple of cii,c2.,•••cAr.m by 11,Efe, J c N , are bounded from below and above. Under the condition (5.18), put 8 = 1 — p =11,E e,—p _ 0. Then, for any disjoint subsets J1, J2, • • • of N , we have by Lemma 5.9 J

1

(5.19)

li mPN ,14`,7=1(1

—

j

j

j

aj „) .

Therefore, to verify p N — > o ( N - .0 ) , it is sufficient to see the existence of disjoint subsets j,, such that EZ-1(3.,,,= co or more simply aj „ c (n = 1 , 2, •••) for

858

Takeshi H irai

a constant c >O. Now, for the term in (5.17), put L k

=

=

{ :1 E J ; i3

k} for k N , and K — { k E N ;

L k * 0 } , If p ={ k E K ;IL k i - P}, then (5.20)

H J E J C iii

k .K (i iE L k c k i) .

The factor H l e L k C k j is the contribution to

the k-th row of (cu).

ILEjci i from i

Therefore

of term s for which

PJ = the sum

Ka2=0.

Since e = 1 and n , e = 1 , we have i

i

f

i

j= th e sum

(5.21)

of terms for which If,2 *O.

Let us evaluate the sum in the right hand side. F or a finite subset Q of and a family (R ),“, of disjoint subsets of Jc N , let ,A ((q; R ), ) be the union of all term s in (5.17) containing the factor FI E (11,-eR c r). Put J'= J\ (U .eR ), then the total sum of elements in ,1((q; R q ) q E Q ) is equal to N

q

q

g

q

e

q

Q

q

q

(by (5.18)) .

H g e Q (11rERq Cq r)'(1 1 ,ep e,)-1 1 q e ( )(11reRg Cq r)

W e denote it by [,..A ((q; R q ) q E Q ) ] . Note th at the term in (5.20) is contained in if and only if Q K and R c L for any q e Q. Let us now first take Q ={ q} , one point set, and R ={ri, r2}, tw o points set. In the sum s2J of all such [.A (q; R ,)]=L A (q; r2})], the monomial (5.20) several ,A(q; R ), and the total number of times of contributes to s2 its contribution is equal to cA ((q ; R q )q E Q )

g

q

g

q

n2=E,,Kk2( ) 2 q

w i t h / =1L l . q

q

N ext take Q ={ q} and R ={ri, r2, r3}, three points set, and let s3J be the sum of a ll su c h [ A (q; R )]. T hen the total num ber of tim es in w hich the monomial (5.20) contributes to sa' is given by g

q

1, 113 =

E q eK 4 3 (

3

) •

Further w e take Q={qi, q2}, two points set, and R „ disjoint tw o points s e ts . Denote by Az the sum of a ll L ii((q; R ) .( )} of such ty p e . T hen the total number of times in which the monomial (5.20) contributes to S 2 is equal q

q

q

R q 2

2

,

to lq i) (1 q , n2,2 — Q T K

Z

w h e r e Q={ qi, q2}.

859

Irreducible unitary representations

In particular, n2,2=0 if 1K,21< 2. N ow let us evaluate m= n2— n3— n2,2 for the m onom ial (5 .2 0 ). In case 1Kk2i =1, let K,2={q}, then m --(

1q

2

1q

)— (

3

)-

1 3

4 (4 -1 )(5 -4 ).

!

Hence m 0 except the cases where / =2, 3, 4 and m=1, 2, 2 correspondingly. In case 1K,21=2, we see similarly that m 0 except the cases w here /q=iL ql= 2 for qE K ,2 and m = 1 a c c o rd in g ly . If 1& 21> 2, th e n m 0 n e c e ssa rily . Thus w e get the following q

Lemma 5 .1 0 . L et s / , 5 3 and 5, 2 be the sum s of L A ((q ; R q )q e )] defined abov e. T hen there holds alw ay s the inequality .

—2

The above evaluation o f 8 , is som etim es not convenient to apply in certain situations. S o w e g iv e an o th er but sim ilar evaluation as follows. Fix a subset I of N . Consider only the terms H j c i n (5.17) or (5.20) with (5.22)

o r " if

./.< 2c /

fo r

then i,E P .

j, j'E J ,

Denote by ,1 1 ((q; f e,),E ) the subset of il( ( q ; R q ) q e Q ) consisting of such terms th a t (5.22) holds, and denote also by s2" s l , i ) the sum of [A '( ( q ; R ) e )] analogous to the sum 5 2 (resp. 53", .42) of [A ( ( q ; 1 ?,) , )]. Since the evaluation of the number of times of contribution of I L . c,„ is always true, we get sim ilarly as above the following Q

q

q

Q

Q

j

Lemma 5 . 1 1 . L et I be a subset and J a finite subset o f N . Then 1 —2

— s3'

.

5 . 6 . Solution of Problem 5.8 (Step 2). sk2 = 0 and so S a 2 i ( s , / / — s 3 i - f ) with 1

i

E C i Ci ,

—

j1

Put /={1. }, one point set, then

,

-

531 -1 =

j2

E

C

ij2 C ii3 •

Since fl i . N d i converges and so di--*1, we m ay assume th a t di fo r iE N . Then S3

11

1

E

3 tii.foc./

E

ci;

—

jE J

Hence 8. 4 - 1 .32' . Now we apply the following 1

— ci;

j 2

2 S2

•

E .i.N c u S 3 1 2

860

Takeshi Hirai

Lemma 5.12. L e t x i, ,x2, •-• b e a f in ite n u m b e r o f non-negativ e real n u m b e rs. Then, 1

E x x j - -tx.olLiX i— m ax{x.i}) 2 j -

0 . Then

(iE

0

A ssum e there ex ists an

U )

as N - œ .

Proof Since d -> 1 as i -> c c , we may assume without loss of generality that di 11< K 13. First take an i = u iE U. F o r this j, we can find a finite subset /lc N such that dz E i e l i c u K O . Then, E , J i c , , d i - K 1 3 .1 -2 K I3 , and by (5.23) we have -

-

2 2 , /, \ 1 1 , --(1. - - K ) - { ( 1 . - - K)—ki — K) } = N 3 '- 8 3 --

2

—

T C )K

(=

IC (put)) . '

we have taken mutually disjoint subsets fi, J2, •••, J , of N such Then we can find a un+1E U and a finite subset \ (U;Lijp) S U C h that, for i =un+i,

Assume

that

(p=1, 2, ••-, n ) .

Tn+IE/V

By a similar evaluation as above, we get Thanks to (5.19), we have p N - (N-■ co).

IC' .

Q.E.D.

By this lemma, we see that it rests for us to check the case where m a x {c ; there exists an injection u of N into itself

j E N } - » 1 as i-> 0 0 . In this case, such that c,,. ( ,) -> 1 (i->œ).

Note that the assumption on c in Probelm 5.8 is symmetric in (i, j). Then, except a case similar to Lemma 5.13, where qN->0, we come to the case -> 1 (j-> 00) with an injection v: N - N , to be checked. Thus, altogether, taking into account e ,= 1 and dz->1, we come to the following three cases, modulo appropriate permutations of rows and columns of c , by elements in u

,

(A)

C

H

* 1;

(B )

1 ; ( C ) ci+N z ,i -> 1;

Irreducible unitary representations

861

0 , where NI >0, N2> O.

as

5 . 7 . Solution of Problem 5.8 (Step 3). Let us first treat the case (A). Then the assumption (5.16') for (c„,) is equivalent to (5.24)

E

00 .

i* j

On the other hand, w e can easily get the inequality sZ,'2' . 2-1 (s2/ J ) 2 , and

therefore S2 —

1

I ,J >( 1

1,1 S 3 —

S 2 ,2 =

J

1

max {c/i0 s2 I i l

W e m a y assume w ith o u t lo ss o f generality ci, obtain (5.25)

j >-

-

1

f j S2 ( 1 - S 2

—4

1 J

I

J

2

' ) .

3/2 fo r i E N . Then, we

)

Solving the inequality x(1— x)

8 , w e get the following - 1

Lemma 5 .1 4 . A ssume for a finite subsets J of N there exists an I cN such that 2+,12

2 - 1 2

(5.26)

s21 '1 < 4 — 4

Then (3.1 ----1—p

•

1/32.

Now we put I = J . Then 1

S2

,1=>

E jcu( Ecu ie e j j

j* i

)

min{ c i i ) • ( E cii) • Ej i,je j i* j

Since c - * 1, we may assume cii 1 / 2 . Hence we have J , J - >

1

S2

( 5 .2 7 )

2 iJEJ j*,

On the other hand, put F = F = J u {p}, one point bigger than / — J . Then the difference — s2" is sm all as shown by S2"

'

S 2

j j

E

E

ie i

1

C I9 jC P P±

E E

iEJ j e j

c p p ) 2 +( d p — c p p ) c p p

cucip

+ max{di} • ( e p —cpp) EJ

862

Takeshi Hirai 5

,3

-

iap- C p p )l e p -

4

Cpp) .

Here d,--E,EN rcz .,5=3/2 and cl1->1 ( i >œ), ep E .isrc,p = 1 , and Cpp *1 (1) >C O ). T his means that s2 ""'-s2 " ..` is sufficiently small if p > 0 . Thus we see from (5.24) and (5.27) that, when we make / - 1 increase one by one appropriately, then some of s21. comes into the interval (5.26), a n d 8j a . 1/32 for such J by Lemma 5.14. From th e above discussion, it is seen by induction that there exists a series of mutually disjoint finite subsets fi, J2, ••• of N such that aj „a 1/32 for n =1 , 2, •••. From this we obtain by (5.19) that p w - 0 as N-+00. Note that, by the symmetry of assumption, we have also qN-1:) in Case (A). -

-

-

-

5 . 8 . Solution of Problem 5 .8 ( S t e p 4 ) . T h e c a s e s ( B ) a n d ( C ) are sim ilar, and so we treat only the form er h e re . In C ase (B ) the condition (5.16') is automatically satisfied. First assume that a similar conditions as (5.24) holds:

E c u = co .

(5.28)

i,je N

Then, we can reduce the situation to Case (A) and get p N .- 0 and q N - - 0 . In fact, for pN, it is sufficient to apply Lemma 5.9 and the result in C ase (A ). For 7 qN, we consider a new (c'i.,) with c'A=_I,k6I+N,czk, (..i-. 2)• P u t q'N for ( 6,,) the quantity similar o qN for (c u ). Then qN -q'N clearly, a n d q'N-)0 since (c . ;.;) is in Case (A). Now assum e (5 .2 8 ) does not h o l d . Then li mN—qN >O. In fact, c1,i+N1) < 0 0 . So let H 16 , and II ie N C i,z + N I converges because E i e N ( d i — us prove p N - - . 0 . Define (c; ; ) as c; C ; d — C z , j+ N i - 1 ( j a 2 ) . Then, qN and therefore we can reduce the case to Ni =1. Assuming N1=1, let us evaluate N . P ut C = (cd ),JE N and let C (N ; k ) be an (N -1 ) x (N -1 ) matrix obtained from C by cutting off i-th row for i a .A1 and j- t h row for j = k and j > N . Further le t DN be a n N x N matrix with ele.

ments d

u

- c i

f

(i< N , j< = N ),

(i< = N ).

Denote by PN (C ) the quantity P N for C = ( c ) . Then we have also PN-1(C(N; k ) ) and pN (D N ), and get by simple calculations dN k

P N (C ) IN (D N )

PN-i(Cuv;k))

H ( E i* k

P N -1 (C (N ; k ))

H

W 5N j* k

Irreducible unitary representations

863

with a constant L > O . T h erefo re P N (C )_ L •

1

C ik E 1sI N E 5i